Few-Body Systems

, Volume 54, Issue 11, pp 2113–2124 | Cite as

Bound States of Energy Dependent Singular Potentials

  • R. Yekken
  • M. Lassaut
  • R. J. Lombard


We consider attractive power-law potentials depending on energy through their coupling constant. These potentials are proportional to 1/|x| m with m ≥ 1 in the D = 1 dimensional space, to 1/r m with m ≥ 2 in the D = 3 dimensional space. We study the ground state of such potentials. First, we show that all singular attractive potentials with an energy dependent coupling constant are bounded from below, contrarily to the usual case. In D = 1, a bound state of finite energy is found with a kind of universality for the eigenvalue and the eigenfunction, which become independent on m for m > 1. We prove the solution to be unique. A similar situation arises for D = 3 for m > 2, except that, in this case, the solution is not directly comparable to a bound state: the wave function, though square integrable, diverges at the origin.


Wave Function Dimensional Space Ground State Energy Attractive Potential Ground State Wave Function 
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  1. 1.
    Shortley G.H.: The Inverse-Cube Central Force Field in Quantum Mechanics. Phys. Rev. 58, 120 (1931)ADSCrossRefGoogle Scholar
  2. 2.
    Case K.M.: Singular Potentials. Phys. Rev. 80, 797 (1950)MathSciNetADSCrossRefMATHGoogle Scholar
  3. 3.
    Guggenheim E.A.: The inverse square potential field. Proc. Phys. Soc. 89, 491 (1966)ADSCrossRefGoogle Scholar
  4. 4.
    Frank W.M., Land D., Spector R.M.: Singular potentials. Rev. Mod. Phys. 43, 36 (1971)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Sen D.: Perturbation theory for singular potentials in quantum mechanics. Int. J. Mod. Phys. A 14, 1789 (1999)ADSCrossRefMATHGoogle Scholar
  6. 6.
    Gupta K.S., Rajeev R.S.: Renormalization in Quantum Mechanics. Phys. Rev. D 48, 5940 (1993)ADSCrossRefGoogle Scholar
  7. 7.
    Camblong H.E., Epele L.N., Fanchiotti H., Garcia Canal C.A.: Renormalization of the Inverse Square Potential. Phys. Rev. Lett. 85, 1590 (2000)ADSCrossRefGoogle Scholar
  8. 8.
    Beane, S.R., Bedaque, P.F., Childress, L., Kryjevsky, A., McGuire, J., van Kolck, U.: Singular potentials and limit cycles. Phys. Rev. A 64, 042103 (2001)Google Scholar
  9. 9.
    Alberg M., Bawin M., Brau F.: Renormalization of the singular attractive 1/r 4 potential. Phys. Rev. A 71, 022108 (2005)ADSCrossRefGoogle Scholar
  10. 10.
    Al-Jaber S.M.: Strongly Singular Potentials in One dimension. An najah Univ. J. Res. (N. Sc.) 19, 167 (2005)Google Scholar
  11. 11.
    Martinez Alonso L.: Schrödinger spectral problems with energy-dependent potentials as sources of nonlinear Hamiltonian evolution equations. J. Math. Phys. 21, 2342 (1980)MathSciNetADSCrossRefMATHGoogle Scholar
  12. 12.
    Martinez Alonso L., Gull Guerrero F.: Modified Hamiltonian systems and canonical transformations arising from the relationship between generalized Zakharov-Shabat and energydependent Schrödinger operators. J. Math. Phys. 22, 2497 (1981)MathSciNetADSCrossRefMATHGoogle Scholar
  13. 13.
    Jaulent M., Jean C.: The inverse s-wave scattering problem for a class of potentials depending on energy. Commun. Math. Phys. 28, 177 (1972)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Formánek J., Lombard R.J., Mareš J.: Wave equations with energy-dependent potentials. Czech. J. Phys. 54, 1143 (2004)CrossRefGoogle Scholar
  15. 15.
    Lombard R.J., Mareš J., Volpe C.: Wave equation with energy-dependent potentials for confined systems. J. Phys. G: Nucl. Part. Phys. 34, 1879 (2007)ADSCrossRefGoogle Scholar
  16. 16.
    Yekken R., Lombard R.J.: Energy-dependent potentials and the problem of the equivalent local potential. J. Phys. A: Math. Theor. 43, 125301 (2010)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Yekken, R., Du spectre au potentiel: Etude de problème inverse dans le cas des états discrets avec extension aux potentiels dépendant de l’énergie, PhD thesis, USTHB, Bab Ezzouar, Alger, Algeria (2009)Google Scholar
  18. 18.
    Garcia-Martinez J., Garcia-Ravelo J., Pena J.J., Schulze-Halberg A.: Exactly solvable energy-dependent potentials. Phys. Lett. A 373, 3619 (2009)MathSciNetADSCrossRefMATHGoogle Scholar
  19. 19.
    Lepage G.P.: Analytic bound-state solutions in a relativistic two-body formalism with applications in muonium and positronium. Phys. Rev. A 16, 863 (1978)ADSCrossRefGoogle Scholar
  20. 20.
    Nieto M.M.: Exact wave-function normalization constants for the B 0 tanh zU 0 cosh−2 z and Pöschl-Teller potentials. Phys. Rev. A 17, 1273 (1978)ADSCrossRefGoogle Scholar
  21. 21.
    Flügge S.: Practical Quantum Mechanics. Springer, Berlin (1994)Google Scholar
  22. 22.
    Haines L.K., Roberts D.H.: One-Dimensional Hydrogen Atom. Amer. J. Phys. 37, 1145 (1969)ADSGoogle Scholar
  23. 23.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 2nd edn. Dover, New York (1972)Google Scholar
  24. 24.
    Nicholson A.F.: Bound States and Scattering in an r −2 Potential. Aust. J. Phys. 15, 174 (1962)ADSCrossRefMATHGoogle Scholar
  25. 25.
    Landau L.D., Lifshitz E.M.: Quantum Mechanics, paragraphs 16 and 35. Pergamon Press, London (1958)Google Scholar
  26. 26.
    Galindo A., Pascual P.: Quantum Mechanics. 2nd edition. Springer, Berlin (1991)MATHGoogle Scholar
  27. 27.
    Newton R.G.: Scattering Theory of Waves and Particles. 2nd edition. Springer, New York (1982)MATHGoogle Scholar
  28. 28.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 2. Mc Graw-Hill, New York (1953)Google Scholar
  29. 29.
    Khuri N.N., Pais A.: Singular Potentials and Peratization. I. Rev. Mod. Phys. 36, 590 (1964)MathSciNetADSCrossRefMATHGoogle Scholar
  30. 30.
    de Llano M., Salazar A., Solis M.A.: Two-dimensional delta potential wells and condensedmatter physics. Rev. Mex. de Fisica 51, 626 (2005)ADSGoogle Scholar
  31. 31.
    Jahan, A., Jafari, M.: Bound state energy of delta-function potential: a new regularization scheme. Iran. J. Sci. Technol. Trans. A 31, 435 (2007)Google Scholar

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© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Faculté de Physique USTHBBab EzzouarAlgeria
  2. 2.Groupe de Physique ThéoriqueInstitut de Physique NucléaireOrsay CedexFrance

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